The spectre of common usage is one of the greatest bugaboos for amateur grammarians, who fear that if we accept a usage because everybody’s using it, we’re weakening the language. For example:

We have often noted that often repeated language and grammar errors seem to become “correct” usage. Wouldn’t it be weird if math used that philosophy? When enough people said 2+2=5, it would! It would still equal 4, of course, but it would also equal 5.

I’ve heard this “2 + 2″ kind of argument many times. It’s a false analogy, a virulent argument that seems reasonable but is wrong at its very core, and is wrong in multiple ways. It misrepresents both language and math, and that makes me mad, because the two things I’ve spent substantial portions of my academic life on are math and language. So let me tell you why this argument is rubbish in both aspects.

Grammaticality isn’t Truth. Math and language are different in a lot of ways. Duh, of course; don’t writers claim to not be “math people” and mathematicians claim to not be “language people”? Well, yeah, but it runs deeper than that. Language comes from our minds and our cultures. There isn’t some true, verifiable, Platonic version of language floating out in the ether that we’re trying to use. It is a social construct. And it’s a nebulous social construct, because we don’t know where it came from, or how it’s evolved over long periods of time. Hell, we don’t even have a clear view of the proper theoretical framework to analyze language (see, e.g., the debates between the Minimalists and HPSG or LFG syntacticans).

If everyone suddenly decides that the word flartish describes an object that is orange-brown, then the word means that. If everyone later decides that flartish describes an object that is wider than it is tall, then the word means that. There is no physical, no Platonic, no real meaning for a word. The meaning of a word is nothing more or less than what the speakers of its language believe it to be. That doesn’t mean that a meaning can’t be wrong; if you start re-assigning definitions to words, like Humpty Dumpty did to glory, you’ll be wrong in that no one will understand you.

The same is true of a language’s grammar. There is no English outside English, nor Telugu outside Telugu. When a language changes — and they do, constantly — that changes what is standard and non-standard in that language. The English you speak now is the results of billions of changes that took place over thousands of years from Proto-Indo-European. The reason that English and Albanian and Urdu aren’t the same language is that they all have undergone changes through common usage. It’s what happens. Math doesn’t change in this same way. Correct proofs can’t become incorrect in the way that grammatical sentences can become ungrammatical.

Sometimes 2+2 doesn’t equal 4. For comparison’s sake, how does mathematical truth work? Well, it works like this.* Before you do anything in math, you have to first lay down a set of axioms, a set of statements that you take as true and cannot prove. The most famous set of these is probably Euclid’s “4 + 1″ postulates of plane geometry, which state the existence of line segments, lines, and circles, as well as the equivalence of all right angles and the uniqueness of parallel lines. If you want to prove something in Euclidean geometry, you build up from those axioms. If you want to define something (a triangle, for instance), you define it in terms of those axioms or in terms of things built up from those axioms. So when you prove something like “the sum of the angles of a triangle is 180 degrees” — one of the rudiments of Euclidean geometry that we learn as kids –, it’s true only as long as the axioms are true.

Now, here’s the rub: a true theorem under one set of axioms is not necessarily true under other axioms. For instance, a triangle has to have 180 degrees worth of angles in plane geometry because Euclid’s axioms hold on a plane. A sphere breaks the Euclidean axioms, though, because parallel lines don’t exist on its surface.** This causes the mathematical truth to become untrue, as shown by a thought experiment, pictured below.

Suppose you decide to go to the North Pole. You start out heading due north from your current position. You get to the North Pole, and take a 90 degree turn to the right, and then head due south until you’re at the same latitude you started at. Now you’re hungry, so you decide to head back to your starting point. So you take another 90 degree turn to the right and head west until you’re back at your starting point. Then you turn 90 degrees one last time to face north and return to your starting alignment.

You're basically Admiral Byrd in this thought experiment, except that you actually made it to the North Pole.

You’ve made a triangle, but you’ve turned a total of 270 degrees. The “true” theorem that a triangle’s angles sum to 180 degrees isn’t true if its axioms aren’t valid. Returning to the seemingly stronger example of 2+2=4, it’s also true under certain arithmetical axioms and groups, but not all. If we’re talking about the group of natural numbers, then yes, 2+2 equals 4. But if we move from base 10 to base 3, 2+2=11. And if we’re talking of the cyclic group of order 3, then 2+2=1, and 4 doesn’t exist.

Mathematical truth is only as true as its underlying axioms, and these examples show that when those axioms are changed, the “truth” falls apart. The claim that common usage could make two plus two not equal four isn’t scary; it’s obvious. We only take this as truth because in common usage, we’re usually talking about the infinite set of natural numbers.

Well, that’s sort of how different languages work. It’s as though English has a rule that says 2+2=7, Malay has a rule that says 2+2=5, and so on. But because the languages have different systems, those different rules can each be valid.

Languages aren’t “right”. There’s a point I really want to hammer home with all of this: who says the English we have now has two and two equalling four? The quote at the beginning of this post presupposes that the present form of English is “right”, and that new deviations from it must therefore be wrong. But our modern English is different not only from other contemporary languages but also from its earlier forms. How do we know that Old English didn’t have two and two equalling four and our modern version has it wrong?

Well, we do know that’s not the case, and that’s because languages aren’t inherently right or wrong. By various measures, one can argue that a specific change to a language brought on by common usage is helpful or harmful, but change itself is not inherently bad — or good, for that matter. So let’s stop deifying the language we currently have and demonizing the changes. It might well be that we’ve got the roles reversed.

—
*: I want to note here that I only have a Bachelor’s degree in math, and we really didn’t go too far into the philosophy of mathematics. I might have some mistakes or controversial interpretations with the details of this section (and if I do, please point them out in the comments), but I believe the core points of this section are accurate.

**: “Lines” on a sphere are restricted to circles that span a diameter of the sphere, so-called “great circles”. Any two great circles on a single sphere will inevitably intersect, so no non-intersecting lines exist in this geometry.

About The Blog

A lot of people make claims about what "good English" is. Much of what they say is flim-flam, and this blog aims to set the record straight. Its goal is to explain the motivations behind the real grammar of English and to debunk ill-founded claims about what is grammatical and what isn't. Somehow, this was enough to garner a favorable mention in the Wall Street Journal.

About Me

I'm Gabe Doyle, currently a postdoctoral scholar in the Language and Cognition Lab at Stanford University. Before that, I got a doctorate in linguistics from UC San Diego and a bachelor's in math from Princeton.

In my research, I look at how humans manage one of their greatest learning achievements: the acquisition of language. I build computational models of how people can learn language with cognitively-general processes and as few presuppositions as possible. Currently, I'm working on models for acquiring phonology and other constraint-based aspects of cognition.

I also examine how we can use large electronic resources, such as Twitter, to learn about how we speak to each other. Some of my recent work uses Twitter to map dialect regions in the United States.

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26 comments

How amazing that English is a living, breathing, evolving creature… all the input that changes it gives it a richness beyond the more-static (and enforced) languages. That being said…. of course there are standards. But balance is a good thing. Thanks for this piece!

‘if you start re-assigning definitions to words’
This conceit is used to disturbing effect in the recent film Dogtooth.

The text in bold near the end struck me, because you’ve used BrE style in the placement of punctuation. Some people might even call it “wrong”. More to the point, the lines you quote at the beginning are so misconceived, that I don’t believe your helpful, reasonable arguments will be persuasive. People become very attached to ideas, even very daft ones.

I must state to begin this remark that I think Plato is overrated. Period. However, (I was an English major), I’m going to print this and keep it ‘forever’ because:
1. I used to hate math but have learned to almost accept that it might be ‘fun’ to be able to prove the correct answers I arrive at by some weird, convoluted path that I can’t explain.
2. It supports my argument with other folks that since cultures are not static (or they die out), neither can be our languages. E.g., is it blood borne or bloodborne, air borne or airborne?
3. Back to math; how can something as apparently abstract as mathematics be learned by any sort of logic or reason if one has nothing to apply it to? (My ‘learning disability’ as related to math which totally evaded me until I could apply it to determining correct medication doses…) Okay, I have two children and four grandchildren for whom math is their prime nemesis. I’m trying to convince them that one can learn math…but I have to back up my contention with proof.
4. The Eureka! moment for me was when my math genius son in law used the illustration of a cooking recipe to help me understand how algebra ‘worked.’ Jeeeeeeesh. I must have missed a day or two of something very important in elementary school that was the entire key to mathematics…wow. Recipes equal formulas for solving those miserable, wretched (yes, I know that’s redundant, but there’s a point being made–I hope) problems that ruined my entire educational process.
5. I liked it.

PS: I lied in the first comment here; I was a nursing major, not an English major. The degree in English is a yet-to-be-achieved dream that I am not working on; too many other things of more importance to me, but I love this site and use it as a portal into the Great World of English Grammar…or something similar.

I bewail the loss of English’s grammar! Where are our dative and accusative noun endings? Why why why did we level all our genitives into the one boring -es, and then drop the -e-, forcing us to cope with the devil’s tool (the apostrophe)? Why have we lost our second person singular pronoun, and our dual pronouns? Why have we gotten so sloppy that we stopped declining “you” and “it” altogether? Why have we jettisoned the lovely strong verbs such as help-holp for that lazy -ed? Why…

Oh, wait? You say that the Golden Age of English was … what, Jane Austen and her singular “they”? Shakespeare and his “thous”? The KJV and its “haths”? Chaucer and his Frenchified borrowings? When, exactly, was the zenith of our tongue? When you were a kid? I getcha….

One time I casually corrected my son and husband – both very strong in math- when they were having a discussion and something was said about triangles having 180 degrees. I asked, “what about triangles with 3 right angles?” Naturally they both very patiently (and maybe a bit patronizingly) explained that such a triangle didn’t exist. I argued a bit just for fun before I got a racquet ball out and drew them a picture. They were impressed with this liberal arts drop out!

One thing I suspect you are missing with your math examples is that the problem lies not with any inconsistency in math, but with the language used when discussing math. Most of us when discussing triangles do not specify that we are speaking only of plane geometry. Likewise we rarely have need to preface our equations with a disclaimer that we are speaking in base ten. I maintain that 2+2 ALWAYS equals 4 (when using whole numbers in base ten).

If you’ll allow me to be way too pedantic… I disagree slightly with your statement that “The meaning of a word is nothing more or less than what the speakers of its language believe it to be.” People can state all sorts of opinions about what they believe words mean, but what’s important is how the words are actually used. For instance, lots of people believe that “decimate” means “destroy ten percent”, but if the word is never used that way, can we say that’s what it means?

I was just reading about American vs British punctuation-within-parenthesis rules.

The gist was that America’s system is intrusive and the British is more “logical”. I know language isn’t logical, but I was swayed by the argument. Why should we muss up verbatim quotes with our own punctuation? Someone said it is a symptom of American Exceptionalism, which I think is a bit ridiculous a claim.

I’ll be using British usage from now on. Maybe I always have. I never really paid attention to the issue before.

Abbie, I love your idea! I believe that I will join you in ‘reverting’ to British punctuation style in all writing where style is left to the author. Fairly often I get into spelling trouble with those extra U’s and the transposed el/le so why not just convert and be done with it eh. Tongue-in-cheek ‘edits’ for my Canadian friends is to put those little U-whiskers into parentheses.

I was so intrigued by the thought experiment that I just pulled out a marker and lip gloss with a spherical lid and drew a triangle. Totally blew my mind. Thanks to this blog, today I have learned about both grammar and math.

And yes, the math analogy is just silly. Language is pretty much arbitrary (except for maybe onomatopoeias). It was invented by people; it exists for us, not vice versa.

Emily: It’s wonderful, isn’t it? Topology brings, well, a whole new dimension to shapes. Re. your last line, I said something similar in a comment on my own blog recently: Language serves us; we do not serve the language.

[…] nearly as gendered a language as the Romance languages, it does lack a gender-neutral option. Attempts to change this, or any parts of the language, have been met with incredulity, derision, hostility, and […]

Language is all but arbitrary. While I’m completely on board with your argument that math has “truths” and language has “usage”, and that different meanings and usages do not have to contradict each other, I want to recommend a passage in “The Tell Tale Brain”, which postulates an idea that the sounds we use to build fundamental words aren’t complete arbitrary. I’m not completely sold on his idea hook, line, and sinker, but he makes an interesting argument.

Mrs Apron, you’re talking about sound symbolism or ideophones. There might be something to this, or there might not. The sounds that are apparently used non-arbitrarily in this manner vary widely across languages.

That’s a good question, perhaps: is there a single phone or phonetic rule which can be found in every human language? I think I’d heard that there isn’t either, and if not, it’d sort of lean against that theory – though change could have eliminated the ur-phone/rule, I suppose.

CaitieCat: It’s been literally decades, but I seem to recall when I was in college one of my linguistics professors said that every language has /ɑ/, /e/, and /o/. However, I don’t know how accurately I’m remembering this, or how accurate the statement was to begin with. As a general rule, every universal rule has exceptions, although the statement that every universal rule has exceptions itself is a universal rule, meaning it may have exceptions. :-)

@dw: Maybe it was /ɑ /, /i/, /u/ and not /ɑ/, /e/, /o/. That would make more sense, since /i/ and /u/ are “at the corners”, so to speak.

You are of course correct about the [əʊ] pronunciation of /o/ (and it can be even more extreme than that, such as the Baltimorese [ɛʊ], but then we get into the issue of whether we’re discussing phonemes or precise phones. Also, for at least some of these speakers, the “proper” [o] still occurs in some instances. For example, I realize /o/ as [əʊ]*, but I realize /ol/ as [o:] My understanding is this is fairly common in the Maryland/Pennsylvania/Delaware area (which is where I grew up), I’ve never listed specifically for it but it wouldn’t surprise me to hear other English speakers realize /ol/ the same way, or maybe as [ol].